Let $M$ be a Kähler manifold. The complex structure on it naturally gives rise to the real analytic structure. I wonder if there exist Kähler manifolds such that the associated symplectic $2$-form $\omega$ is $C^\infty$-smooth but not real analytic.
The answer is positive for any Kähler manifold.
Consider first surfaces. Take a compact Riemann surface $\Sigma$, then any symplectic form on it is associated to a Kähler form, so the answer is yes, since there are plenty of non-analytic $2$-forms.
More generally, for any Kähler manifold $M$ take any Kähler form $\omega$ and let $\omega_1$ be a closed $(1,1)$-form supported in a ball $B\subset M$. Then $\omega+\varepsilon \omega_1$ is Kähler for small $\varepsilon$, but not analytic.
Not quite an answer, but something related to it.
There is this paper where the authors prove, that for every symplectic manifold $(M,\omega)$ there is an analytical manifold $M^a$ and an analytical symplectic form $\omega^a$ such that $(M,\omega)$ and $(M^a,\omega^a)$ are symplectomorphic.
Edit: The manifold $(M^a,\omega^a)$ is unique up to symplectomorphisms.